diff --git a/education/math/MATH1210 (calc 1)/Integrals.md b/education/math/MATH1210 (calc 1)/Integrals.md index 0486832..6e7c1c1 100644 --- a/education/math/MATH1210 (calc 1)/Integrals.md +++ b/education/math/MATH1210 (calc 1)/Integrals.md @@ -119,3 +119,20 @@ This theorem tells us that a continuous function on the closed interval will obt - $\int \dfrac{1}{u\sqrt{u^2 - a^2}} du = \dfrac{1}{a}arcsec(\dfrac{|u|}{a}) + C$ # Length of a Curve +## Review of the Mean Value Theorem +If $f$ is a continuous function on the interval $[a, b]$ and differentiable on $(a, b)$, then there exists a number $c$ in the interval $(a, b)$ such that: + +$$ f'(c) = \dfrac{f(b) - f(a)}{b - a} $$ + +This also implies that for some $c$ in the interval $(a, b)$: +$$ f(b) - f(a) = f'(c)(b-a) $$ + +## Intuitive Approach +Given that we divide a curve into $n$ sub-intervals, and we can find the location of the right endpoint of each interval. + +With a series of points on a curve we can find the distance between each point. + +As we increase $n$, the precision of which the curve is estimated increases. + +This means that: +$$ len(curve) = \lim_{n \to \infty} \sum_{i=1}^{n}(\p{}$$ \ No newline at end of file